456 CHAPTER 9 Inferences from Two Samples Inferences About Two Means: Independent Samples Objectives 1. Hypothesis Test: Conduct a hypothesis test of a claim about two independent population means. 2. Confidence Interval: Construct a confidence interval estimate of the difference between two independent population means. Notation For population 1 we let m1 = population mean x1 = sample mean s1 = population standard deviation s1 = sample standard deviation n1 = size of the first sample The corresponding notations m2, s2, x2, s2, and n2, apply to population 2. Requirements KEY ELEMENTS 1. The values of s1 and s2 are unknown and we do not assume that they are equal. 2. The two samples are independent. 3. Both samples are simple random samples. 4. Either or both of these conditions are satisfied: The two sample sizes are both large (with n1 7 30 and n2 7 30) or both samples come from populations having normal distributions. (The methods used here are robust against departures from normality, so for small samples, the normality requirement is loose in the sense that the procedures perform well as long as there are no outliers and departures from normality are not too extreme.) (If the third requirement is not satisfied, alternatives for hypothesis tests include the resampling methods of bootstrapping and randomization described in Section 9-5, or the Wilcoxon rank-sum test described in Section 13-4.) Hypothesis Test Statistic for Two Means: Independent Samples (with H0: M1 = M2) t = 1x1 - x22 - 1m1 - m22 Bs2 1 n1 + s2 2 n2 1where m1 - m2 is often assumed to be 02 Degrees of Freedom When finding critical values or P-values, use the following for determining the number of degrees of freedom, denoted by df. (Although these two methods typically result in different numbers of degrees of freedom, the conclusion of a hypothesis test is rarely affected by the choice.) 1. Use this simple and conservative estimate: df = smaller of n1 − 1 andn2 − 1 2. Technologies typically use the more accurate but more difficult estimate given in Formula 9-1. FORMULA 9-1 df = 1A + B22 A2 n1 - 1 + B2 n2 - 1 where A = s2 1 n1 and B = s2 2 n2 Note: Answers in Appendix D include technology answers based on Formula 9-1 along with “Table” answers based on using Table A-3 with the simple estimate of df given in option 1 at the left. P-Values: P-values are automatically provided by technology. If technology is not available, refer to the t distribution in Table A-3. Use the procedure summarized in Figure 8-3 on page 380. Critical Values: Refer to the t distribution in Table A-3.
RkJQdWJsaXNoZXIy NjM5ODQ=